What is the inverse of the function $g(x)=-2(x-4)$ ? $g^{-1}(x)=$
Solution: Let's start by replacing $g(x)$ with $y$. $y=-2(x-4)$ If a function contains the point $(a,b)$, the inverse of that function contains the point $(b,a)$. So if we swap the position of $x$ and $y$ in the equation, we get the inverse relationship. In this case, the function is $y=-2(x-4)$, so the inverse relationship is $x=-2(y-4)$. Solving this equation for $y$ will give us an expression for $g^{-1}(x)$. $\begin{aligned} x&=-2(y-4)\\\\ -\dfrac{1}{2}x&=y-4\\\\ -\dfrac{1}{2}x+4&=y\\\\\\ \end{aligned}$ The inverse of the function is $g^{-1}(x)=-\dfrac{1}{2}x+4$. [I saw someone solve this problem by originally solving for x. Were they wrong?]